Theorem 2moswapv 2631

Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap 2646 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 10-Apr-2004.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexexvw 2630. (Revised by Wolf Lammen, 2-Oct-2023.)

Assertion

Ref	Expression
2moswapv	⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2moswapv

Step	Hyp	Ref	Expression
1		nfe1 2147	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	nfmov 2560	. . . 4 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
3		nfe1 2147	. . . . 5 ⊢ Ⅎ𝑥∃𝑥(∃𝑦𝜑 ∧ 𝜑)
4	3	nfmov 2560	. . . 4 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)
5	1, 2, 4	moexexlem 2628	. . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
6	5	expcom 414	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)))
7		19.8a 2174	. . . . 5 ⊢ (𝜑 → ∃𝑦𝜑)
8	7	pm4.71ri 561	. . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑))
9	8	exbii 1850	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑))
10	9	mobii 2548	. 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
11	6, 10	syl6ibr 251	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537  ∃wex 1782  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-mo 2540

This theorem is referenced by:  2euswapv  2632  2rmoswap  3696